When \(\Sigma \)  is a surface of finite type, \(\operatorname{Map} (\Sigma )\)  is finitely generated.
A finite generating set defines a word metric on \(\operatorname{Map} (\Sigma )\)  which is well-defined
up to quasi-isometry independent of the particular finite generating set. The large
scale geometry of the mapping class group, that is the geometry of the quasi-isometry class of
such metrics, has been studied extensively [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
